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I am given the definition of a hyperplane in $M$-dimensional space as the locus of points $\vec{d}\in\mathbb{R}^M$ with $\vec{w}\neq\vec{0}$ and $\theta\in\mathbb{R}$ satisfying

$$\sum_{i=1}^{M}w_{i}d_{i}=\theta$$

and asked to state the "geometric significance" of $\theta$ in the cases where $M=2,3$. Using the definition of the dot product, I can easily conclude that

$$\theta=\|\vec{w}\|\|\vec{d}\|\cos{\phi}$$

where $\phi$ is the angle between $\vec{w}$ and $\vec{d}$. Now, since $\|\vec{d}\|\cos{\phi}$ is nothing but the length of the orthogonal projection of $\vec{d}$ onto $\vec{w}$, I suppose I should conclude $\theta$ to be the product of that projection length by the length of $\vec{w}$. However, that doesn't strike me as a particularly "geometric" interpretation, and certainly does not provide any insight into the 2- and 3-dimensional cases.

How can I (re-)formulate my thoughts into something which clearly shows the geometric significance of $\theta$? (Disclosure: this is a homework problem, so I'm merely looking for a hint provided I'm not just overlooking something silly.)

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    If you fix $\vec w$ and $\theta$, then the points $\vec d$ will all be those whose orthogonal projections onto $\vec w$ are $\theta$. I.e. if you were to extend $\{\vec w\}$ to an orthogonal basis of $\Bbb R^M$, all your points $\vec d$ would have the same first coordinate with respect to that basis -- but all the other coordinates could vary unrestricted. Draw a picture in $2$d and confirm that this gives you a unique line. – 2017-02-21
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    I can see that - it's actually a following part to the same question to demonstrate the same. I was more interested in the direct relationship between that line and the parameter $\theta$ in a geometrical sense. For example, if it were given that $\|\vec{w}\|=1$, I would state that $\theta$ is the distance from the origin to that line. Alas, I cannot take that assumption. – 2017-02-21
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    Well, $\left|\dfrac{\theta}{\|\vec w\|}\right|$ is the distance from the origin to that line. – 2017-02-21
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    True. (And I forgot that $\theta$ is a signed distance w.r.t. $\vec{w}$, so thank you for implicitly reminding me.) But is there nothing more to say? I'll be a little let down by this question if I spent hours on a wild goose chase! (Also, I'll accept your first response as an answer if you'll submit it as one.) – 2017-02-21

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The hyperplane (thought of as a set of points) is perpendicular to the line $\operatorname{span}(\vec w)$ (note, the vectors $\vec d$ that point to the hyperplane are not perpendicular to $\vec w$) and it is a distance $\left|\dfrac{\theta}{\|\vec w\|}\right|$ from the origin at its closest point. You can even say that that closest point is along $\operatorname{span}(\vec w)$, in the direction of $\vec w$ if $\theta>0$ and in the opposite direction if $\theta<0$.

Maybe you can phrase that better because my phrasing looks like it might be a little confusing.